We give a partial answer to a question of Carlitz asking for a closed formula for the number of distinct representations of an integer in the Fibonacci base.

We prove that a word of length n from a finitely ambiguous context-free language can be generated at random under uniform distribution in O(n2 log n) time by a probabilistic random access machine assuming a logarithmic cost criterion. We also show that the same problem can be solved in polynomial time for every language accepted by a polynomial time 1-NAuxPDA with polynomially bounded ambiguity.

We introduce the notion of a k-synchronized sequence, where k is an integer larger than 1. Roughly speaking, a sequence of natural numbers is said to be k-synchronized if its graph is represented, in base k, by a right synchronized rational relation. This is an intermediate notion between k-automatic and k-regular sequences. Indeed, we show that the class of k-automatic sequences is equal to the class of bounded k-synchronized sequences and that the class of k-synchronized sequences is strictly contained in that of k-regular sequences. Moreover, we show that equality of factors in a k-synchronized sequence is represented, in base k, by a right synchronized rational relation. This result allows us to prove that the separator sequence of a k-synchronized sequence is a k-synchronized sequence, too. This generalizes a previous result of Garel, concerning k-regularity of the separator sequences of sequences generated by iterating a uniform circular morphism.

This contribution extends the notions of roots and periodicity to strings of transfinite lengths. It shows that given a transfinite string, either it possesses a unique root or the set of its roots are equivalent in a strong way.

We say that two languages X and Y are conjugates if they satisfy the conjugacy equationXZ = ZY for some language Z. We study several problems associated with this equation. For example, we characterize all sets which are conjugated via a two-element biprefix set Z, as well as all two-element sets which are conjugates.

In this paper we introduce a sharpening of the Parikh mapping and investigate its basic properties. The new mapping is based on square matrices of a certain form. The classical Parikh vector appears in such a matrix as the second diagonal. However, the matrix product gives more information about a word than the Parikh vector. We characterize the matrix products and establish also an interesting interconnection between mirror images of words and inverses of .

In this paper methods and results related to the notion of minimal forbidden words are applied to the fragment assembly problem. The fragment assembly problem can be formulated, in its simplest form, as follows: reconstruct a word w from a given set I of substrings (fragments) of a word w. We introduce an hypothesis involving the set of fragments I and the maximal length m(w) of the minimal forbidden factors of w. Such hypothesis allows us to reconstruct uniquely the word w from the set I in linear time. We prove also that, if w is a word randomly generated by a memoryless source with identical symbol probabilities, m(w) is logarithmic with respect to the size of w. This result shows that our reconstruction algorithm is suited to approach the above problem in several practical applications e.g. in the case of DNA sequences.

This paper surveys the area of Free Burnside Semigroups. The theory of these semigroups, as is the case for groups, is far from being completely known. For semigroups, the most impressive results were obtained in the last 10 years. In this paper we give priority to the mathematical treatment of the problem and do not stress too much neither motivation nor the historical aspects. No proofs are presented in this paper, but we tried to give as many examples as was possible.

In a previous paper, the authors studied the polynomial closure of a variety of languages and gave an algebraic counterpart, in terms of Mal'cev products, of this operation. They also formulated a conjecture about the algebraic counterpart of the boolean closure of the polynomial closure – this operation corresponds to passing to the upper level in any concatenation hierarchy. Although this conjecture is probably true in some particular cases, we give a counterexample in the general case. Another counterexample, of a different nature, was independently given recently by Steinberg. Taking these two counterexamples into account, we propose a modified version of our conjecture and some supporting evidence for that new formulation. We show in particular that a solution to our new conjecture would give a solution of the decidability of the levels 2 of the Straubing–Thérien hierarchy and of the dot-depth hierarchy. Consequences for the other levels are also discussed.